# Gravity-induced viscous flow around a cylinder

## Homework Statement

A viscous liquid with density and viscosity ##\rho## and ##\mu## respectively is discharged onto the upper surface of a cylinder with radius ##a## at a volume flow rate ##Q##. This is a gravity-driven flow, and it forms a film around the cylinder--see picture.

What is the thickness $h$ of the layer as a function ##\theta##, ##a##, ##\rho##, ##\mu##, ##g##, and ##Q##.

Navier-Stokes
Continuity

## The Attempt at a Solution

First off, let's assume the flow is incompressible and viscous-dominated, newtonian, steady, 2-D, and that no pressure gradient is present. Then continuity and Navier-Stokes equations are $$\nabla \cdot \vec{V} = 0$$ and $$g\hat{j} = \nu \nabla^2 \vec{V}$$
Now if we adopt a cylindrical coordinate system where ##x=rcos\theta## and ##y=r\sin\theta## we have
$$\frac{1}{r}\frac{\partial (r u_r)}{\partial r}+\frac{1}{r}\frac{\partial u_\theta}{\partial \theta}=0$$ and
$$g(\sin\theta \hat{e_r}+\cos\theta \hat{e_\theta})=\nu \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial \vec{V} }{\partial r} \right)+\frac{1}{r^2} \frac{\partial^2 \vec{V}}{\partial \theta^2}$$

where ##\vec{V} = u_\theta \hat{e_\theta}+u_r\hat{e_r}##. Boundary conditions would be no slip along the surface, ##\vec{V} = 0## at ##r=a##. Another would be no stress along the surface of the thin film. And lastly we would have some incoming velocity related to ##Q##, which would be the velocity at ##r=a,\theta=0##. Any ideas how to proceed?

Thanks so much!

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Chestermiller
Mentor
I would start out by looking at flow down an inclined plane of constant angle, and seeing where that takes me. It's got to be a pretty good approximation for most angles on the cylinder.

Chet

Thanks for the response Chet! I did this but how can you extrapolate this to a cylinder?

Nidum
Gold Member